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Full text of "Mathematical And Physical Papers - Iii"

28 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS 15. The form of the equations of condition (30) points out sin2<9 as a factor of P, and since the operation sin 6 -^ -r^ -^ performed on the function sin20 reproduces the same function with a coefficient - 2; it will be possible to satisfy equations (27) and (28) on the supposition that sin2# is a factor of ^ and ^2#. Assume then Patting for convenience n*f^l=p'm*.; ......................... (32), and substituting in (27) and (28), we get /1»-^/1W = 0 ........... . ............ (33), /2»-^/2W-m2/2(r} = 0 .................. (34). The equations of condition (30), (31) become, on putting /(r) (35), (36). We may obtain p from (29) by putting for ^ its value (r), replacing after differentiation 2/1(r) by its equiva- * When this operation is performed on the function sin0 dYJdO, the function is reproduced with a coefficient - i (i + 1). F4 here denotes a Laplace's function of the itk order, which contains only one variable angle, namely 6. Now \f/ may be ex- panded in a series of quantities of the general form sm6dYi/d6. For, since we are only concerned with the differential coefficients of ^ with respect to r and 0, we have a right to suppose i// to vanish at whatever point of space we please. Let then ;//=0 when r=a and 0 = 0. To find the value of ^ at a distance r from the origin, along the axis of x positive, it will be sufficient to put 0 = 0, dd = Q in (20), and integrate from r — a to r, whence i// = 0. To find the value of i// at the same dis- tance r along the axis of x negative, it will be sufficient to leave r constant, and integrate d$ from. 0 = 0 to 0 = 7T. Referring to (26), we see that the integral vanishes, since the total flux across the surface of the sphere whose radius is r must be equal to zero. Hence ^ vanishes when 0 = 0 or =TT, and it appears from (26) that when 6 is very small or very nearly equal to x, ^ varies ultimately as sin2 0 for given values of r. and t. Hence ^ cosec 0, and therefore f\j/ cosec 6 dd, is finite even when sin 0 vanishes, and therefore/^ cosec 0 dd may be expanded in a series of Laplace's functions, and therefore i// itself in a series of quantities of the form sin0 It was somewhat in this way that I first obtained the form of the function ^.